First-principles studies of beryllium doping of GaN

Transcription

1 PHYSICAL REVIEW B, VOLUME 63, First-principles studies of beryllium doping of GaN Chris G. Van de Walle * and Sukit Limpijumnong Xerox Palo Alto Research Center, 3333 Coyote Hill Road, Palo Alto, California 9434 Jörg Neugebauer Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D Berlin-Dahlem, Germany Received 12 October 2; published 8 June 21 The structural and electronic properties of beryllium substitutional acceptors and interstitial donors in GaN are investigated using first-principles calculations based on pseudopotentials and density-functional theory. In p-type GaN, Be interstitials, which act as donors, have formation energies comparable to that of substitutional Be on the Ga site, which is an acceptor. In thermodynamic equilibrium, incorporation of Be interstitials will therefore result in severe compensation. To investigate the kinetics of Be interstitial incorporation and outdiffusion we have explored the total-energy surface. The diffusivity of Be interstitials is highly anisotropic, with a migration barrier in planes perpendicular to the c axis of 1.2 ev, while the barrier for motion along the c axis is 2.9 ev. We have also studied complex formation between interstitial donors and substitutional acceptors, and between hydrogen and substitutional beryllium. The results for wurtzite GaN are compared with those for the zinc-blende phase. Consequences for p-type doping using Be acceptors are discussed. DOI: 1.113/PhysRevB PACS number s : y, 66.3.Jt, Eq, 72.8.Ey I. INTRODUCTION Nitride semiconductors are currently being used in a wide variety of electronic and optoelectronic devices. Controllable doping is essential for all of these devices. For n-type doping of nitrides, a number of elements e.g., Si can be successfully used as dopants, and carrier concentrations exceeding cm 3 can routinely be achieved. 1 The situation is less favorable for p-type doping. Magnesium is the acceptor of choice; it can be incorporated in concentrations up to about 1 2 cm 3, but because of its large ionization energy 21 mev, Ref. 1 the resulting room-temperature hole concentration is only about 1 18 cm 3, i.e., only about 1% of Mg atoms are ionized at room temperature. Increasing the Mg concentration beyond 1 2 cm 3 leads to a saturation and decrease in the hole concentration. 2 In previous work, we have proposed that the Mg solubility limit is the main cause of this behavior. 3 The limited conductivity of p-type doped layers constitutes an impediment for progress in device applications. It would be desirable to identify an alternative acceptor that would exhibit higher solubility and/or lower ionization energy. Previous computational studies 4 6 have addressed a variety of candidate acceptors, including Li, Na, K, Be, Zn, Ca, and Cd. Only Be emerged as a viable acceptor, exhibiting higher solubility and lower ionization energy than Mg. However, it also emerged that self-compensation may occur due to incorporation of Be on interstitial sites, where it acts as a donor. 5 The conclusions of the previous theoretical work seem promising enough, however, to warrant a more comprehensive investigation, which is the subject of the present paper. Experimentally, Be doping of GaN in MBE molecular 7 15 beam epitaxy has been reported by various groups. These experimental studies have involved both wurtzite and cubic phases of GaN; the cubic phase can be obtained by growth on cubic substrates, under appropriately tailored conditions. Another technique for Be incorporation that has been attempted is ion implantation. 16,17 No electrical conductivity results were reported; characterization was mostly by optical spectroscopy. A photoluminescence PL peak just below the band edge of GaN was found and attributed to Be. Assuming that this PL line results from a Be substitutional acceptor, an acceptor ionization energy can be extracted but this has resulted in a wide range of values, ranging from 9 to 25 mev, depending on the assumptions made in the analysis. Brandt et al. reported that using oxygen as a codopant along with Be results in high p-type conductivity. 18,19 Yamamoto and Katayama-Yoshida 2 proposed an expanation for these codoping results in terms of Be Ga -O N -Be Ga complexes. The purpose of the present study is to investigate the behavior of Be in GaN in detail, with particular focus on the diffusion properties of interstitial Be. Indeed, a thorough understanding of the stability and diffusivity of interstitial Be will allow us to assess the likelihood that interstitial Be will be incorporated during growth and if it is, whether it might be removed during a subsequent annealing procedure. To address this issue, we have mapped out the total-energy surface for an interstitial Be atom moving through the GaN lattice; this enables us to extract the barriers for Be diffusion. In addition, we have studied the interaction between interstitial Be donors (Be int ) and substitutional Be acceptors (Be Ga ). Coulomb attraction leads to formation of a complex with a large binding energy 1.35 ev. Because hydrogen is known to play an important role in p-type doping of GaN, we have also investigated its interaction with Be acceptors, leading to the formation of Be-H complexes with a binding energy of 1.81 ev. We have also performed additional investigations in order to make contact with the above-mentioned experimental studies. Because the experimental studies have involved both wurtzite and cubic phases of GaN, we have performed most of our calculations for both phases. The PL results prompted us to examine the optical ionization energy of the Be accep /21/63 24 / /$ The American Physical Society

2 VAN de WALLE, LIMPIJUMNONG, AND NEUGEBAUER PHYSICAL REVIEW B tor in detail, and we found a large Franck-Condon shift due to an unusual atomic relaxation of the acceptor in its neutral charge state. Because of the interest in codoping with oxygen, finally, we have performed comprehensive calculations for Be Ga -O N and Be Ga -O N -Be Ga complexes. The organization of this paper is as follows. In Sec. II, we discuss the computational approach and methods. Section III lists our results for isolated impurities, and Sec. IV contains results for complexes. Section V, finally, contains a discussion and comparison with existing experiments. Section VI summarizes the paper and contains suggestions for future experimental work. II. THEORETICAL APPROACH A. Pseudopotential density-functional theory We use the density-functional theory 24 in the local density approximation LDA and ab initio norm-conserving pseudopotentials, with a plane-wave basis set. 25 Since the Ga 3d electrons play an important role for the chemical bonding in GaN, they cannot simply be treated as core electrons. Explicitly taking Ga 3d as valence electrons is one way to obtain more realistic results, as discussed in detail in Ref. 26. However, the localized nature of the Ga 3d states significantly increases the computational demand, requiring an energy cutoff of at least 6 Ry. We have therefore used the so-called nonlinear core correction, 27 with an energy cutoff of 4 Ry. We have tested this approximation by comparing formation energies for interstitial Be with results obtained using explicit inclusion of Ga 3d electrons as valence electrons; the values agree to within.1 ev. B. Supercells The wurtzite WZ structure is the lowest-energy structure of GaN, but the zinc-blende ZB structure is only slightly higher in energy ( 1 mev per two-atom unit; Ref. 28 and can be obtained by epitaxial growth on a zinc-blende substrate. We have therefore performed calculations for Be in both the zinc-blende and the wurtzite structures. As we will see, results for substitutional Be are very similar in both phases, but the properties of interstitial Be are quite different. All impurity calculations are carried out at the theoretical lattice constant, in order to avoid any spurious relaxations. For wurtzite GaN, we find a th 3.89 Å compared with a expt 3.19 Å ). The calculated c/a ratio is experiment 1.627, very close to the ideal c/a ratio of 8/3. For zinc-blende GaN, we find a th 4.37 Å, which is to within.1 Å 2 larger than the wurtzite lattice constant. In order to study the atomic and electronic structure of an impurity in the GaN crystal, we construct an artificial unit cell supercell composed of several primitive GaN unit cells and containing one impurity. The larger the supercell size, the closer our results will be to the case of a single, isolated impurity, because interactions between impurities in neighboring supercells are suppressed. The computation time, however, increases as N 2 ln N where N is the number of atoms in the supercell. It is therefore imperative to choose the smallest possible cell size that produces reliable results FIG. 1. Top view along 1 direction of the GaN wurtzite structure: small circles represent nitrogen, large circles gallium. The shaded area corresponds to the primitive unit cell. The translation vectors for the primitive unit cell and for the 32-, 72-, and 96-atom supercells are also shown. for the problem under investigation. We have performed calculations for several different supercell sizes, in the process checking convergence. For the wurtzite structure, we have employed 32-, 72-, and 96-atom supercells. The 32-atom supercell is composed of eight wurtzite GaN primitive unit cells each containing four atoms, such that each translation vector of the supercell is doubled from that of the basic unit cell. If we choose the lattice translation vectors for wurtzite GaN to be a 1 (a,,), a 2 ( a/2, 3a/2,), and a 3 (,,c) see Fig. 1, the translation vectors for the 32-atom supercell are defined as a i (32) 2a i, where i 1,2,3. For the 72-atom supercell, the translation vectors are a 1 (72) 3a 1, a 2 (72) 3a 2, and a 3 (72) 2a 1. As can be noted from Fig. 1, both 32- and 72- atom supercells suffer from the problem that the separation between impurities in neighboring supercells imagined to be located at the center of the supercell is quite different when measured along different directions. We have therefore considered a 96-atom supercell that has translation vectors that are mutually perpendicular, leading to a cell with orthorhombic symmetry. The translation vectors of this 96-atom supercell are a 1 (96) 4a 1 2a 2, a 2 (96) 3a 2, and a 3 (96) 2a 3. For the zinc-blende structure, we have used 32- and 64- atom supercells, which have been extensively discussed in the context of point-defect calculations in zinc-blende semiconductors. The 32-atom supercell has bcc symmetry, while the 64-atom cell is cubic, consisting of the conventional eight-atom cubic unit cell of the zinc-blende structure doubled in each direction. Within the supercell, we allow relaxation of several shells of host atoms around the impurity. For wurtzite, 24 host atoms were relaxed in the 32-atom cells. In the 96-atom cell, at least 46 host atoms plus the impurity were allowed to relax, including all atoms within 4.8 Å of any impurity. For a substitutional impurity, this corresponds to seven shells of atoms. In the zinc-blende 64-atom cell, the same relaxation radius leads to at least 44 atoms being relaxed five shells. Results of our convergence tests will be discussed in Sec. III A. The tests indicate that, for zinc blende, 32-atom and 64-atom supercells yield very similar results, indicating convergence. For wurtzite, the absolute values of formation energies are not yet converged in a 32-atom cell. The 96-atom

3 FIRST-PRINCIPLES STUDIES OF BERYLLIUM... PHYSICAL REVIEW B cell results are expected to be converged for substitional impurities, these results are very close to those for the ZB 64-atom cells. However, 32-atom wurtzite cell calculations are very useful to obtain energy differences between different positions of the impurity in the lattice; we found these energy differences to be within.3 ev of those calculated in the 96-atom cell. Since mapping out the total-energy surface describing the motion of interstitial Be through the lattice requires a large number of calculations, we have used the 32-atom cell for that purpose. C. Special k points Brillouin-zone integrations were carried out following the Monkhorst-Pack scheme 29 with a regularly spaced mesh of n n n points in the reciprocal unit cell shifted from the origin to avoid picking up the point as one of the sampling points. Symmetry reduces this set to a set of points in the irreducible part of the Brillouin zone. Convergence tests indicate that for zinc blende the sampling yields total energies that are converged to better than.1 ev in both 32-atom and 64-atom supercells. For the 32-atom wurtzite supercell with Be 2 int, a set does not yield fully converged results; energy differences are obtained reliably, however. We also compared the total energies calculated using and sets. The former leads to only one irreducible k point while the latter leads to three irreducible k points. Although the absolute value of the total energy differs significantly between the two calculations, the energy differences between different Be int locations change by less than.1 ev. The calculations required to explore the totalenergy surface were therefore carried out using only one special k point. In the 96-atom wurtzite cell, finally, the k-point mesh produces fully converged results. As we will see in the next section, the formation energy of non-neutral impurities takes into account that electrons are exchanged with the Fermi level. The Fermi level E F is referenced to the valence-band maximum in the bulk, i.e., E F at the top of the valence band (E v ) in bulk GaN. We obtain E v by calculating the position of the valence-band maximum in GaN with respect to the average electrostatic potential. We need to bear in mind, however, that the average electrostatic potential in our defect supercells is not the same as that in bulk GaN; indeed, our first-principles calculations for periodic structures which fill all space do not provide an absolute reference for the electrostatic potential, due to the long-range nature of the Coulomb potential. An explicit alignment procedure is thus needed between the electrostatic potential in the defect supercell and the electrostatic potential in the bulk. This problem is similar to that of calculating heterojunction band offsets, and similar techniques can be used to address these issues. 31 Here we have chosen to align the electrostatic potentials by inspecting the potential in the supercell far from the impurity and aligning it with the electrostatic potential in bulk GaN. This leads to a shift in the reference level, V, which needs to be added to E v in order to obtain the correct alignment. The resulting shifts are taken into account in our expressions for formation energies in the next section. Interstitial Be atoms in GaN are double donors. In this work we have focused on the 2 charge state (Be 2 int ), but we make some comments about other charge states in Sec. IIIC2. Finally, we have also studied the (Be Ga -Be int ) complex; from the charge states of the interstitial and substitutional species, it can be inferred that this complex has a charge state of 1. The neutral charge state is discussed in Sec. IV A. D. Charge states Beryllium substituting on the Ga site in GaN is a single acceptor; we therefore study two charge states, Be Ga and Be Ga. In the case of a shallow acceptor such as Be Ga the level introduced in the Kohn-Sham band structure due to the presence of the impurity is merely a perturbation of the host band structure. This acceptor level therefore exhibits essentially the same dispersion as the uppermost valence band. In the negative charge state, the acceptor level is filled but in the neutral charge state, one electron is removed from this level. For a true, isolated acceptor corresponding to a calculation in a very large supercell, the electron would be removed from the top of the valence band, at the point. But in our finite-size supercells the electron is actually taken out of the highest occupied Kohn-Sham level at the special k points, where the band energy is lower than at the point. A correction is therefore needed, obtained from the energy differences between the highest occupied state at the point and the special k points. The magnitude of this correction ranges from.2 ev in the 96-atom cell to.5 ev in the 32-atom cells. This correction term, which we call E corr and which has been defined as a positive number, has been taken into account in all our results for neutral acceptors. 3 E. Formation energies The formation energy E f determines the concentration c of the impurity in the semiconductor, through the expression c N sites exp E f /kt, where N sites is the number of sites in the lattice per unit volume where the impurity can be incorporated, k is Boltzmann s constant, and T is the temperature. Equation 1 in principle holds only in thermodynamic equilibrium. Growth of a semiconductor is obviously a nonequilibrium process; however, many growth environments involve high enough temperatures to ensure reasonable mobility of impurities and defects at or near the surface, establishing conditions that approximate equilibrium. Furthermore, even in nonequilibrium conditions the formation energy of an impurity or defect is a useful concept, since configurations with high formation energies will obviously be difficult to incorporate or will attempt to evolve to a lower-energy state. The formation energies of Be 2 int, Be Ga, Be Ga, and (Be Ga -Be int ) are defined as

4 VAN de WALLE, LIMPIJUMNONG, AND NEUGEBAUER PHYSICAL REVIEW B E f Be 2 int E tot Be 2 int E tot GaN, bulk Be 2 E F E v V Be int, E f Be Ga E tot Be Ga E tot GaN, bulk Be Ga E corr, E f Be Ga E tot Be Ga E tot GaN, bulk Be Ga E F E v V Be Ga, E f Be Ga -Be int E tot Be Ga -Be int E tot GaN, bulk 2 Be Ga E F E v V Be Ga -Be int. 5 E tot X is the total energy derived from a supercell calculation with one impurity X in the cell, and E tot GaN, bulk is the total energy for the equivalent supercell containing only bulk GaN. Ga and Be are the chemical potentials of Ga and Be, respectively. E F is the Fermi level, referenced to the valence-band maximum in the bulk. Due to the choice of this reference, we need to explicitly include the energy of the bulk valence-band maximum, E v, in our expressions for formation energies of charged states. As discussed in Sec. II D, we also need to add a correction term V to align the reference potential in our defect supercell with that in the bulk. Finally, the correction term E corr that appears in the formation energy of Be Ga was discussed in Sec. II D. The chemical potentials depend on the experimental growth conditions, which can be either Ga rich or N rich. For the Ga-rich case, we use Ga Ga[bulk] to place an upper limit on Ga ; this places a lower limit on N, calculated from N min E tot GaN Ga, where E tot GaN is the total energy of a two-atom unit of bulk GaN, calculated for the structurally optimized wurtzite structure. For the N-rich case, the upper limit on N is given by N N[N2 ], i.e., the energy of N in a N 2 molecule at T ; this yields a lower limit on Ga calculated from min Ga E tot GaN N. The total energy of GaN can also be expressed as E tot GaN Ga[bulk] N[N2 ] H f GaN, where H f GaN is the enthalpy of formation, which is negative for a stable compound. Our calculated value for H f GaN is.48 ev expt ev; Ref. 32. By combining Eq. 6 or Eq. 7 with Eq. 8, we observe that the host chemical potentials vary over a range corresponding to the magnitude of the enthalpy of formation of GaN. For Be, the upper bound on the chemical potential arises from the solubility-limiting phase, which is Be 3 N 2. There is no lower bound on Be, minus infinity corresponding to the total absence of Be from the growth environment. Equilibrium with Be 3 N 2 implies Be 2 N 3 Be[bulk] 2 N[N2 ] H f Be 3 N 2, where H f Be 3 N 2 is the calculated enthalpy of formation of Be 3 N 2, which is 5.5 ev. 33 The experimental value of H f Be 3 N 2 is 6.11 ev. 34 Equation 9 allows us to relate Be to N, assuming equilibrium with Be 3 N 2. Combining this information with the expression for the formation energy of Be Ga Eq. 3, we find that the lowest formation energy and hence the highest concentration of Be Ga, i.e., the solubility limit will occur under nitrogen-rich conditions, i.e., when N N[N2 ]. 35 Ultimately, one wants to calculate concentrations of native defects or impurities. For that purpose, one would choose a particular set of atomic chemical potentials corresponding to the growth conditions, and substitute the formation energies from Eqs. 2 5 in the expression for concentrations, Eq. 1. This still leaves the Fermi level as a free parameter. This remaining variable is fixed by imposing the condition of charge neutrality: the total charge resulting from charged defects or impurities, along with the charge due to any free carriers in valence and conduction bands, needs to be equal to zero. This leads to a unique position for the Fermi level. It turns out to be very informative to explicitly plot the dependence of formation energies on Fermi levels: this immediately provides insight into the electrical activity donor or acceptor character, and shows whether certain defects will act as compensating centers. We will therefore present our results in this format. F. Thermal and optical ionization energies The thermal ionization energy E A of the Be Ga substitutional acceptor is defined as the transition energy between the neutral and negative charge states of the dopant sometimes the notation (/ ) is also used for this quantity. This transition energy is defined as the Fermi-level position where the formation energies of these two charge states are equal, i.e., E f Be Ga E F E A E f Be Ga. 1 From Eqs. 3 and 4, it then follows that E A E f Be Ga E F E f Be Ga 11 E tot Be Ga E tot Be Ga E corr E v V Be Ga. 12 For purposes of defining the thermal ionization energy, it is implied that for each charge state the atomic structure is relaxed to its equilibrium configuration. The atomic positions in these equilibrium configurations are not necessarily the same for both charge states. Indeed, we will see that they are quite different for the neutral and negative charge states of Be Ga. This difference leads to an interesting and sizable effect when the ionization energy of the dopant is determined optically. Assume the following simplified picture of a photoluminescence experiment. The exciting light creates electron-hole pairs. The holes can be trapped at Be Ga centers,

5 FIRST-PRINCIPLES STUDIES OF BERYLLIUM... PHYSICAL REVIEW B FIG. 2. Schematic configuration coordinate diagram illustrating the difference between thermal and optical ionization energies. The curve for Be Ga is vertically displaced from that for Be Ga assuming the presence of an electron in the conduction band. E rel is the Franck-Condon shift, i.e., the relaxation energy that can be gained, in the negative charge state, by relaxing from configuration q equilibrium configuration for the neutral charge state to configuration q equilibrium configuration for the negative charge state. turning them into Be Ga. Using our definition of the thermal ionization energy E A, the equilibrium configuration of the Be Ga e state where e is an electron at the bottom of the conduction band is E g E A higher than the equilibrium configuration of Be Ga, where E g is the band gap. Electrons in the conduction band can then recombine with the hole on the acceptor, as illustrated in Fig. 2. This leads to emission of a photon with energy E PL. During this emission process, the atomic configuration of the acceptor remains fixed i.e., in the final state, the acceptor is in the negative charge state, but with a structure configuration coordinate q ) that is the same as that for the neutral charge state. The difference between the energy of this configuration and that of the equilibrium configuration q is the relaxation energy E rel the Franck-Condon shift. From Fig. 2 it is clear that E PL E g opt E A E rel. If the optical ionization energy E A is defined as the energy difference between the band gap and the PL line, we find that E opt A E g E PL E A E rel. This simplified picture ignores excitonic effects, etc., but it does show that the ionization energy extracted from an optical measurement should be larger than the thermal ionization energy E A by an amount E rel. If the atomic configuration in the two charge states is significantly different as will turn out to be the case for Be Ga ), E rel can be sizable. III. RESULTS FOR ISOLATED IMPURITIES FIG. 3. Calculated defect formation energies as a function of Fermi level for various configurations of Be impurities in wurtzite GaN, under a Ga-rich conditions and b N-rich conditions. A. The substitutional Be Ga acceptor in zinc-blende and wurtzite GaN The formation energies as a function of E F are shown in Fig. 3 for Ga-rich and N-rich conditions. As discussed in Sec. II E, N-rich conditions favor the incorporation of Be on Ga sites, leading to the largest possible Be Ga concentration the solubility limit. Moving toward Ga-rich conditions does not alter our qualitative conclusions; numerical results for any choice of the chemical potentials can always be obtained, of course, by referring to the expressions for formation energies in Eqs In particular, by combining these expressions with the information on chemical potentials in Eqs. 8 and 9, we find that by moving from N-rich to Ga-rich conditions the formation energy of Be Ga increases by 1 3 H f GaN, while the formation energy of Be int decreases by 2 3 H f GaN. Consequences of these results for optimizing growth conditions will be discussed in Sec. V. The formation energies are also summarized in Table I; the numbers given there assume the Fermi level is located at the top of the valence band. We notice that, for the substitu- TABLE I. Calculated formation energies for Be substitutional acceptors (Be Ga ), Be interstitial donors (Be int ), and Be Ga -Be int complexes in wurtzite WZ and zinc-blende ZB GaN. The formation energies see Eqs. 2 5 are given for E F, i.e., E F located at the top of the valence band, and for nitrogen-rich conditions. The dependence of formation energies on Fermi level is shown in Fig. 3. Be Ga Be Ga 2 Be int Formation energies ev WZ ZB (Be Ga -Be int )

6 VAN de WALLE, LIMPIJUMNONG, AND NEUGEBAUER PHYSICAL REVIEW B TABLE II. Relaxations around Be substitutional acceptors (Be Ga ) in wurtzite WZ and zincblende ZB GaN. d Be denotes the displacement of the Be atom from its nominal lattice site, expressed as a percentage of the bulk Ga-N bond length. d and d denote the percentage of change in the Be-N bond length, again expressed referenced to the bulk Ga-N bond length. For wurtzite, the symbol denotes the direction parallel to the c axis 1 ; the other bond directions are denoted by the symbol. For zinc blende, the symmetry around the impurity is lowered to C 3v, with denoting the particular 111 direction in which the impurity is displaced, and denoting the other 111 directions. Be Ga Be Ga WZ ZB WZ ZB d Be.3% % 1.7% 11.4% d 4.6% 4.9% 12.2% 14.5% d 5.4% 4.9% 8.7% 9.3% tional acceptor Be Ga, the results for wurtzite and zinc-blende structures are remarkably similar, the differences being smaller than the calculational error bars. The ionization energies, as defined in Eq. 11, are.17 ev for the wurtzite structure and.2 ev for zinc blende. Table II summarizes our results for the atomic configuration of substitutional Be in wurtzite and zinc-blende GaN. The atomic configurations of Be Ga and Be Ga in wurtzite GaN are also schematically illustrated in Fig. 4. This figure, as well as all subsequent schematic representations of atomic structures, is based on calculated atomic coordinates obtained from a 96-atom supercell calculation. The substitutional Be atom is surrounded by four N neighbors. A contraction of the Be-N bond length is expected since the atomic radius of Be is smaller than that of Ga: the covalent radius of Be is.9 Å, while that of Ga is 1.26 Å. 36 Based on this difference, and using a bond length of 1.95 Å for bulk GaN, FIG. 4. Schematic representation of atomic positions in the (112 ) plane for a Be Ga and b Be Ga in wurtzite GaN. Large circles represent Ga atoms, medium circles N atoms, and the hatched circle represents Be. Dashed circles indicate ideal atomic positions, dashed lines bonds in the ideal lattice. The numbers denote the percentage of change in the bond lengths, referenced to the bulk Ga-N bond length as in Table II. we would expect a contraction of the Be-N bond length by 18% compared to the Ga-N bond length. This is indeed observed in the compound Be 3 N 2, where the bond lengths are Å, % shorter than in GaN. Using ionic radii Å for Be 2 and.47 Å for Ga 3 ), we would expect a contraction of about 1%. The actual magnitude of the contraction of the Be-N bond length as calculated for a Be Ga impurity in GaN is less than 6%. The Be-N bond length is thus somewhat longer than is optimal for Be-N, reflecting the fact that the energy cost associated with moving the surrounding N atoms as well as further shells of atoms is quite large. As discussed in Ref. 5, this makes substitutional Be in GaN less energetically favorably than could be hoped based on the large Be-N bond strength. In spite of this unfavorable size match, it was observed that the formation energy of Be Ga compares favorably with that of other substitutional acceptors. These observations about size mismatch do show, however, that strategies aimed at relieving the lattice strain associated with incorporating substitutional Be could have an important impact. 38 In the negative charge state, the Be atom is located essentially on the substitutional lattice site. In the neutral charge state the situation is very different. Here the Be atom undergoes a sizable relaxation off the nominal lattice site, by more than 1% of the bond length. We also note that the d bond length becomes very large, indicating that the bond with one of the N neighbors is significantly weakened. Simultaneously, the bonds with the other three N neighbors contract by almost 1%. The large off-center displacement of Be Ga was found in calculations for supercells of different sizes and shapes, and occurs in both the wurtzite and the zinc-blende structures see Table II. In the wurtzite structure, the configuration listed in Table II and depicted in Fig. 4 involves the Be atom moving along the c axis. The configuration in which Be Ga moves along the direction of one of the other Be-N bonds is very close in energy within.3 ev, with very similar relaxations around the Be atom. This tendency of Be to form stronger bonds with three of its nitrogen neighbors is not unexpected. Consider, for instance, the atomic structure of the hexagonal modification of Be 3 N 2 Ref. 39 : some of the Be atoms are surrounded by a deformed tetrahedron of N atoms, while others are surrounded by a triangle of nitrogen atoms. It thus seems to be energetically quite acceptable for Be to form strong bonds with only three of its four N neighbors. This tendency will also emerge when we inspect bonding configurations for interstitial Be. Since the equilibrium configuration for the neutral acceptor is so different from that for the negative charge state, we expect the relaxation energy E rel as defined in Sec. II F and Fig. 2 to be quite sizable. E rel is the energy difference, for Be Ga, between the atomic configuration that yields the minimum energy for the neutral charge state i.e., with large lattice relaxation and the atomic configuration that yields the minimum energy for the negative charge state i.e., with Be on the ideal lattice site ; this yields E rel.1 ev. Following the discussion in Sec. II F, we thus expect the ionization energy of the acceptor as determined optically to be signifi

7 FIRST-PRINCIPLES STUDIES OF BERYLLIUM... PHYSICAL REVIEW B cantly higher than the thermal ionization energy. Consequences of this result for experimental observations will be discussed in Sec. V B. The large off-center relaxation of the Be acceptor at least in the neutral charge state is reminiscent of the large lattice relaxations experienced by donor atoms in the so-called DX centers; 4 in this process, the shallow donor (d ) is converted into a deep acceptor (DX ). A similar mechanism for acceptors would involve placing the acceptor in a positive charge state (AX ) and letting it undergo a large lattice relaxation. We have carefully examined whether such a mechanism can occur for Be, but have found no evidence for stability of a Be Ga configuration. We have examined the stability of Be Ga by displacing the Be atom from the substitutional site; such a displacement would be the first step in a process where the Be would move away from the substitutional site, forming a Be interstitial and leaving a Ga vacancy behind. We investigated various configurations in which a Be interstitial was placed in the vicinity of a Ga vacancy. When the overall system was placed in a negative charge state corresponding to the Be Ga reference state, the system immediately relaxed back, without any barrier, to the Be Ga configuration. For other charge states, we found it impossible to obtain converged calculations for the system with Be moved off the substitutional site, indicating the instability of such configurations. Although it is relatively easy to form Be interstitials see Sec. III C, the energy cost to create a Ga vacancy is prohibitively high, particularly in p-type GaN; 41 it is therefore exceedingly unlikely that substitutional Be would leave the Ga site. B. Interstitial Be Be int in zinc-blende GaN The formation energy of the interstitial species Be 2 int in zinc-blende GaN is listed in Table I. This formation energy is negative. One should keep in mind, however, that the formation energy depends on the Fermi-level position see Eq. 2 ; the value given in Table I is for E F, i.e., E F positioned at the valence-band maximum. The formation energy increases and becomes positive when the Fermi level is higher in the band gap see Fig. 3. As described in Sec. II E, the final Fermi-level position is determined by charge neutrality, and this position will always correspond to positive formation energies. Table I shows that the formation energy is higher in the wurtzite structure, though still low enough to be a concern for compensation, as discussed in Sec. III C and Sec. V. The global minimum for Be 2 int in ZB GaN is at the tetrahedral interstitial site surrounded by nitrogen atoms (T N d ). The Be atom is essentially at the center of the cage, and the relaxation of the N atoms is such that the Be-N distance is approximately 1.73 Å; the N atoms thus move toward Be by more than.2 Å. Interstitial diffusion in the zinc-blende structure proceeds via a path through hexagonal rings, with an intermediate position at the tetrahedral interstitial site surrounded by Ga atoms (T Ga d ). The energy difference between T Ga d and T N d can thus serve as an estimate for the diffusion activation energy; this value is equal to 3.74 ev. We conclude that interstitial diffusion of Be has a high barrier in zinc-blende GaN. C. Interstitial Be Be int in wurtzite GaN The formation energy of the interstitial species Be 2 int is shown in Fig. 3 and listed in Table I. Figure 3 indicates that the formation energy of the interstitial donor is low enough to be a serious concern for compensation. We already pointed out that this concern is even more severe in the zincblende structure, where the formation energy of the interstitial is.61 ev lower. When trying to grow p-type GaN doped with Be acceptors, one should thus be aware of potential compensation by Be interstitials. The extent to which interstitials form a problem, as well as the extent to which their incorporation can be controlled, depend sensitively on their diffusivity. The fact that Be interstitials can be readily incorporated is due to the small atomic size of the Be atom; one might therefore suspect that this small atom could also diffuse readily through the lattice. 1. Total-energy surfaces To study the behavior of the Be interstitial in GaN, we have calculated the total energy of Be int at various locations in GaN. For each position of Be, the surrounding host atoms are allowed to relax. The resulting energy values as a function of the coordinates of the Be position, R Be, define the energy surface: E E(R Be ). The energy surface is therefore a function of three spatial dimensions. In order to obtain accurate results, a large data base of energy values for a large number of spatial coordinates is needed. In order to render the computational effort tractable, we carried out these calculations in 32-atom wurtzite cells with one k point. As discussed in Sec. II C, this affects energy differences between different Be sites by less than.4 ev. This accuracy is sufficient not to affect the qualitative shape of the energy surface. Local minima and saddle points identified in this investigation were subsequently also calculated using the 96- atom supercell. We calculated 59 different locations in the irreducible portion of the wurtzite unit cell. These locations were not equally spaced in the crystal, because we chose to calculate more points near the important locations the local minima and saddle points. For each local minimum, we also allowed Be int to relax to ensure that the actual minimum energy and lattice structure were obtained. In order to analyze and visualize the total-energy surface, we want to make optimal use of symmetry information. 42 Since the energy surface is a function of R Be, it possesses the full symmetry of the crystal. To make effective use of these symmetries, an analytic description of the surface is essential. We achieve this through expansion in a basis set with the appropriate symmetry. We then perform a fitting of the total-energy results to this set of basis functions using a least-squares method. As basis functions, we use symmetrized plane waves that reflect the full symmetry of the wurtzite crystal. 42 We also make use of the physical fact that

8 VAN de WALLE, LIMPIJUMNONG, AND NEUGEBAUER PHYSICAL REVIEW B FIG. 5. Schematic representation of atomic positions in the (112 ) plane of wurtzite GaN. The large circles represent Ga atoms, medium circles N atoms. The high-symmetry interstitial sites are indicated: O is the octahedral interstitial site and T the tetrahedral interstitial site. O indicates the position of the global minimum for Be 2 int. the Be impurity can never approach any host atom too closely exchange processes, which would carry a very high energy cost, are not included in the total-energy surface. To this end, we add some ad hoc high-energy values near the host atoms to the calculated data set. This addition does not affect the shape of the energy surface in the relevant regions away from the atoms. Since the energy surface is a function of three spatial dimensions, it is difficult to present the results in a single plot. For visualization purposes, we need to show a cut of the energy surface, restricting the Be coordinates to a single plane. Judicious choice of such planes ensures that we convey all the essential information, i.e., stable and metastable sites as well as barriers between them. For the system considered here, the (112 ) plane see Fig. 5 and the 1 plane perpendicular to the c axis are the relevant planes. The energy surface can then be displayed as a contour plot or as a perspective plot of the energy along the z axis as a function of coordinates in the plane. In the wurtzite structure, there are two distinct types of interstitial open space, as shown in Fig 5. T is the tetrahedral interstititial site or cage site. This site is equidistant from four Ga and four N atoms. O is the octahedral interstitial site or tunnel site. This site is in the interstitial channel along the c axis. The O site is equidistant from six Ga and six N atoms. Both sites are obvious candidates for local minima or, more generally, extrema for Be int. 2. Be int near the O site Placing the Be interstitial at the O site and allowing all atoms to relax results in a position which we call the O site where Be optimizes its distance to its N neighbors. This position constitutes the global minimum in the total-energy surface for Be 2 int in GaN. As can be seen in Fig. 6 a, the Be atom stays along the axis of the open channel, and moves close to the plane of the N atoms. The resulting Be-N distance 1.59 Å is close to the optimum Be-N bond length as observed in, say, Be 3 N 2. In the process, the N atoms move toward Be by.22 Å. The formation energy at this global minimum is.61 ev higher than for Be in ZB GaN see Table I. One might think that the cause for the higher formation energy in wurtzite is the smaller size of the interstitial cage. However, the fact that for Be int at the O site the nitrogen neighbors actually move toward the interstitial indicates that available volume is not an issue. The Be atom clearly likes to bind to the N atoms; in the open channel of the WZ structure, it can only effectively form bonds with three N atoms. At the T N d site in ZB, on the other hand, it can strongly bind to four N neighbors. We suggest that this is the reason for the lower formation energy in zinc-blende GaN. We also find that the bonding between the Be and N atoms exhibits a significant covalent component, in addition to the ionic component; this emerges from an inspection of the charge density corresponding to various eigenstates throughout the valence band. For this global minimum, we have also examined other charge states. The Be interstitial does not introduce any Kohn-Sham states within the LDA band gap, indicating that it behaves as a shallow donor. We can calculate the formation energy of the 1 and charge states by placing one or two electrons in the lowest unoccupied level. As for the case of shallow acceptors, the formation energies need to be corrected with a term E corr reflecting the energy difference between electrons at the special point and at the point, as discussed in Sec. II D. In addition, these formation energies suffer from the LDA band gap error; however, if we focus on energy differences between charge states, as appropriate for the calculation of ionization energies, then the results do not directly depend on the LDA error if we express the ionization level with respect to the conduction-band minimum. Following this procedure, we find the / transition to correspond to an effective-mass state, while the / level is slightly deeper, namely,.9 ev below the conduction band. We will revisit this result in Sec. V B. Coming back to Be 2 int, the features of the energy surface in the neighborhood of the O site can be obtained from Fig. 7. This figure includes a contour plot Fig. 7 a and a perspective plot Fig. 7 b for the (112 ) plane that cuts through the hexagonal channel. We discuss these plots in more detail in the following sections. 3. Be int at the T site We place Be int at the ideal T site and allow the host atoms to relax. If we allow Be int to relax along 1, we find that it stays almost at the ideal position of the T site; we therefore do not need to introduce a separate notation for the minimum-energy site, as we did in the case of the O site. The T site is thus a local minimum for displacements parallel to 1. However, when we allow displacements perpendicular to 1 and move Be slightly off this axis, in order to break the symmetry, we find that the T site is a local maximum in the plane perpendicular to 1. We conclude that T is actually a saddle point. A contour plot and a perspective plot for the (112 ) plane that cuts through the T site

9 FIRST-PRINCIPLES STUDIES OF BERYLLIUM... PHYSICAL REVIEW B FIG. 6. Schematic representation of atomic positions in the (112 ) plane for Be 2 int at various interstitial sites in wurtzite GaN. Large circles represent Ga atoms, medium circles N atoms, and the hatched circle represents Be. Dashed circles indicate ideal atomic positions, dashed lines bonds in the ideal lattice. Distances between Be and its neighbors are given in Å and changes in Ga-N bond lengths are given as the percentage change from the bulk Ga-N bond lengths. The panels correspond to a Be interstitial in the following configurations: a O site; b T site; c A site; and d the Be Ga -Be int complex. are shown in Figs. 7 a and 7 b, respectively. The corresponding plots for the 1 plane that cuts through the T site are shown in Figs. 7 c and 7 d. Figure 7 d clearly shows that T is a local maximum in the 1 plane. In the (112 ) plane, the T site is a local minimum in the 1 direction and a local maximum in the 11 direction. The position of the Be atom and the relaxations of the host atoms are shown in Fig. 6 b. Although the host atoms undergo quite large relaxations, the Be atom is squeezed rather tightly between the Ga and N atoms along the c axis, with a Be-Ga distance of 1.92 Å and a Be-N distance of 1.54 Å. In spite of this tight fit, the T site is only 1.18 ev higher in energy than the global minimum at the O site as calculated in the 96-atom supercell; in the 32-atom cell, the energy difference is.9 ev. The energy at T is probably lowered due to the fact that the Be atom can now bond with four N neighbors; but the size constraints, in particular the repulsion with the Ga atom along the c axis, raise the energy. 4. Be int at the A site In the process of exploring the total-energy surface for Be int, we discovered an additional local minimum. Figures 7 a and 7 b show a contour plot and a perspective plot for

10 VAN de WALLE, LIMPIJUMNONG, AND NEUGEBAUER PHYSICAL REVIEW B FIG. 7. Color Calculated total-energy surfaces for Be interstitials in GaN. a Contour plot and b perspective plot of the total-energy surface for Be 2 int in a (112 ) plane through the host atoms as defined in Fig. 5. In the contour plot, the interval is.5 ev and the real-space dimensions along both directions are in Å. c Contour plot and d perspective plot of the total-energy surface for Be 2 int in a 1 plane through the T site. This 1 plane thus also cuts through the middle of Ga-N bonds along the c axis. In the contour plot, the interval is.25 ev and the real-space dimensions along both directions are in Å. Dashed circles show the projections of the atomic positions of the host atoms and dashed lines show the projections of Ga-N bonds. the (112 ) plane that cuts through both T site and O site. We notice that there is a local minimum located between the T site and the O site; we will call this site the A site. The atomic positions for Be int at the A site are shown in Fig. 6 c. This local minimum has a formation energy only.7 ev higher than the global minimum as calculated in the 96- atom supercell; in the 32-atom cell, the energy difference is.3 ev. Again, the driving force seems to be the formation of Be-N bonds; two of the N atoms move toward Be by.19å, while a third moves by.8 Å. This results in three Be-N bonds with bond lengths of 1.6 Å. Simultaneously, a Ga atom that would prevent Be from approaching N moves

11 FIRST-PRINCIPLES STUDIES OF BERYLLIUM... PHYSICAL REVIEW B FIG. 8. Total energy of Be 2 int at positions along the center of the open channel in WZ GaN, measured from the O site, and referenced to the energy at the global minimum (O ). Solid circles: the results from k-mesh calculations; open circles: k-mesh calculations; solid line: interpolation curve. away, undergoing a displacement of almost 2% of the Ga-N bond length. Still, the resulting Be-Ga distance of 2.2 Å is probably the determining factor limiting the stability of this site. 5. Diffusion path for Be int The total-energy surfaces allow us to study the diffusion path of the Be interstitial along different directions. We focused on the 2 charge state, which is the relevant charge state for Be int at its global minimum see Sec. III C 2. Inspection of the electronic structure for various positions of Be along its migration path indicates that there is no tendency for localized electronic levels to move into the band gap, and hence no possibility for charge-state changes to occur. To study the diffusion path along the 1 direction, we first calculated the formation energies for Be int at several positions along a 1 direction through an O site i.e., along the center of the open channel. The results are shown in Fig. 8. This diffusion path yields a diffusion barrier of 3.1 ev in the 32-atom cell, very close to the 3. ev obtained in a 96-atom cell. Inspection of the full total-energy surface, as shown in Fig. 7, indicates that the lowest-energy path does not run exactly along the center of the channel. The dashed line shows the diffusion path along the center of the channel, while the dotted line shows the actual path; the barrier along the latter path is only slightly lower, at 2.9 ev. For diffusion in a direction perpendicular to the c axis, the interstitial travels between O sites in adjacent channels via a path that has a saddle point at the T site. The diffusion barrier is thus equal to the energy difference between the T and O sites, which is 1.18 ev in the 96-atom cell;.9 ev in the 32-atom cell. We thus find a very large difference in barrier heights for diffusion parallel to 1 vs perpendicular to 1. A rough estimate of the temperature at which the interstitials would be mobile can be obtained by taking the usual definition of an activation temperature, i.e., the temperature at which the jump rate is one per second, and assuming a prefactor of 1 13 s 1, i.e., a typical phonon frequency. With a 1.18 ev barrier, the Be interstitial should be mobile in directions perpendicular to the c axis at temperatures of about 18 C. Diffusion along the c axis, on the other hand, with a barrier of 2.9 ev, would require temperatures of about 85 C. The large anisotropy of the diffusion barriers can be attributed to the features of Be bonding with the host atoms that we described above: Be 2 int prefers to be close to at least three N atoms, with a bond length comparable to that in Be 3 N 2. At the same time, Be 2 int tries to stay as far away as possible from the Ga atoms, with which it has a repulsive interaction. As can be seen from Fig. 8, the saddle point for migration along 1 occurs close to the plane of the Ga atoms. These atoms form part of a hexagonal ring in the chair configuration; in this configuration, the center of the triangle formed by the three N atoms is displaced from the center of the triangle formed by the Ga atoms by.6 Å along the c axis. Because of the Be-Ga repulsion, squeezing through the triangle of Ga atoms carries a large energy cost. Simultaneously, the Be atom is far away from the N atoms with which it prefers to bond. These features conspire to create a large barrier for Be motion along the c axis. For motion perpendicular to the c axis, the Be atom also has to squeeze through hexagonal rings, but this time these rings have the boat configuration. In this configuration, the center of the triangle formed by the N atoms coincides with the center of the triangle formed by the Ga atoms. This configuration thus allows the Be to stay more closely bonded to the N atoms, even while squeezing through the triangle of Ga atoms, resulting in a lower barrier. IV. RESULTS FOR COMPLEXES A. Complexes between interstitial Be and substitutional Be Be int -Be Ga in wurtzite GaN Since interstitial Be is a donor (Be 2 int ), we expect it will bind with substitutional Be, which is an acceptor (Be Ga ). We explored various possible atomic configurations for the complex; the final stable geometry is illustrated in Fig. 6 d. We notice a similarity with the A site configuration of the Be interstitial, ilustrated in Fig. 6 c. Indeed, although this metastable site for Be 2 int was somewhat higher in energy than the O site, the Be interstitial is quite close to a substitutional Ga site in this A site configuration. This offers the possibility for forming a strong bond to a Be Ga substituting on this site. The relaxation can be described as follows. Be int moves toward Be Ga while Be Ga moves away slightly in the opposite direction. The two Be atoms form a bond with a bond length of 1.94 Å. An alternative way of looking at this configuration is as a split interstitial, with the two Be atoms sharing the substitutional Ga site; the center of the Be-Be bond is displaced.46 Å from the ideal Ga position. Both Be atoms